Simplify and expand the following expression: $ \dfrac{5}{5n + 50}- \dfrac{2}{2n - 12}- \dfrac{2}{n^2 + 4n - 60} $
First find a common denominator by finding the least common multiple of the denominators. Try factoring the denominators. We can factor a $5$ out of denominator in the first term: $ \dfrac{5}{5n + 50} = \dfrac{5}{5(n + 10)}$ We can factor a $2$ out of denominator in the second term: $ \dfrac{2}{2n - 12} = \dfrac{2}{2(n - 6)}$ We can factor the quadratic in the third term: $ \dfrac{2}{n^2 + 4n - 60} = \dfrac{2}{(n + 10)(n - 6)}$ Now we have: $ \dfrac{5}{5(n + 10)}- \dfrac{2}{2(n - 6)}- \dfrac{2}{(n + 10)(n - 6)} $ The least common multiple of the denominators is: $ 10(n + 10)(n - 6)$ In order to get the first term over $10(n + 10)(n - 6)$ , multiply by $\dfrac{2(n - 6)}{2(n - 6)}$ $ \dfrac{5}{5(n + 10)} \times \dfrac{2(n - 6)}{2(n - 6)} = \dfrac{10(n - 6)}{10(n + 10)(n - 6)} $ In order to get the second term over $10(n + 10)(n - 6)$ , multiply by $\dfrac{5(n + 10)}{5(n + 10)}$ $ \dfrac{2}{2(n - 6)} \times \dfrac{5(n + 10)}{5(n + 10)} = \dfrac{10(n + 10)}{10(n + 10)(n - 6)} $ In order to get the third term over $10(n + 10)(n - 6)$ , multiply by $\dfrac{10}{10}$ $ \dfrac{2}{(n + 10)(n - 6)} \times \dfrac{10}{10} = \dfrac{20}{10(n + 10)(n - 6)} $ Now we have: $ \dfrac{10(n - 6)}{10(n + 10)(n - 6)} - \dfrac{10(n + 10)}{10(n + 10)(n - 6)} - \dfrac{20}{10(n + 10)(n - 6)} $ $ = \dfrac{ 10(n - 6) - 10(n + 10) - 20} {10(n + 10)(n - 6)} $ Expand: $ = \dfrac{10n - 60 - 10n - 100 - 20}{10n^2 + 40n - 600} $ $ = \dfrac{-180}{10n^2 + 40n - 600}$ Simplify: $ = \dfrac{-18}{n^2 + 4n - 60}$